Let be the angle between the planes and . Let be the line that meets at the point and makes an angle with the normal of . If is the angle between and , the n is equal to:

9 Explanation: Solution Step 1: Find the angle between the planes The angle between two planes is the same as the angle between their normal vectors and . From the equations: The formula for the cosine of the angle between them is: Step 2: Determine the angle According to the problem: The line makes an angle with the normal of . is the angle between the line and the plane . By geometric definition, the angle between a line and a plane ( ) and the angle between that same line and the plane's normal ( ) are complementary: Step 3: Calculate the final value We need to find the value of : Multiplying the results: Final Answer: The value is 9 .

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JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $\theta$ be the angle between the planes $P_1 : \vec{r} \cdot (\hat{i} + \hat{j} + 2\hat{k}) = 9$ and $P_2 : \vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 15$ . Let $L$ be the line that meets $P_2$ at the point $(4, -2, 5)$ and makes an angle $\theta$ with the normal of $P_2$ . If $\alpha$ is the angle between $L$ and $P_2$ , then $(\tan^2 \theta)(\cot^2 \alpha)$ is equal to:

JEE 2023 — Mathematics PYQ

Choose the correct answer:

Explanation

Solution

Step 1: Find the angle $\theta$ between the planes

Step 2: Determine the angle $\alpha$

Step 3: Calculate the final value

Explanation

Solution

Step 1: Find the angle $\theta$ between the planes

Step 2: Determine the angle $\alpha$

Step 3: Calculate the final value